Extending Friedmann Equations Using Fractional Derivatives Using a Last Step Modification Technique: The Case of a Matter Dominated Accelerated Expanding Universe

Abstract

We present a toy model for extending the Friedmann equations of relativistic cosmology using fractional derivatives. We do this by replacing the integer derivatives, in a few well-known cosmological results with fractional derivatives leaving their order as a free parameter. All this with the intention to explain the current observed acceleration of the Universe. We apply the Last Step Modification technique of fractional calculus to construct some useful fractional equations of cosmology. The fits of the unknown fractional derivative order and the fractional cosmographic parameters to SN Ia data shows that this simple construction can explain the current accelerated expansion of the Universe without the use of a dark energy component with a MOND-like behaviour using Milgrom’s acceleration constant which sheds light into to the non-necessity of a dark matter component as well.

Classical and quantum gravity with fractional operators

“…As an application, under the hypothesis that the matter distribution of galaxies behaves as a fractal medium with non-integer dimension, solving a Poisson equation with fractional Laplacian (−Δ) γ one can describe the properties of such matter distribution with a fractional version of Newtonian gravity and account for the observed galaxy rotation curves without invoking dark matter, if the fractional exponent γ is close to 3/2 [36, 37] 2 • The ordinary time derivatives of the Friedmann equations of homogeneous and isotropic cosmology were replaced by fractional derivatives in [47][48][49][50]. In particular, in [50] it was shown that supernovae data on the late-time acceleration of the universe can be explained by fractional Friedmann equations with derivative order close to 3/2. These equations were not obtained from the symmetry reduction of any background-independent non-linear gravitational action.…”

Fractional cosmic strings

“…On the contrary, LSM heavily relies on GR: the switch from partial to fractional derivative is performed on the Euler-Lagrange equations obtained from the usual Einstein-Hilbert action. This is by far the oldest and the most manageable method, and as such it has been used to extend Friedman-Robertson-Walker cosmology [33] and the Schwarzschild exterior solution [15]. ISM is an in-between approach: in the case of the fractional action-like variational approach, the switch from partial to fractional derivative is applied to the Einstein-Hilbert action before this latter is extremized (see for example equations ( 20) and ( 33) in [31]).…”

Fractional-order derivatives in cosmological models of accelerated expansion

“…In recent paper [65], a toy model for extending the Friedmann equations of relativistic cosmology with fractional derivatives is considered. The cosmological consequences of replacing the integer time derivatives in the Friedmann equations with fractional derivatives (that is using LSM) are explore.…”

“…The authors of Ref. [65] then apply a fitting procedure to the SN Ia data to estimate the unknown order values of the fractional derivative and fractional cosmographic parameters. It is noted that such a simple approach could explain the current accelerated expansion of the universe without using the dark energy component.…”